Subharmonicity of the variations of K\"ahler-Einstein metrics on pseudoconvex domains

0.21 MB·English

by  Young-Jun Choi

SUBHARMONICITY OF VARIATIONS OF KA¨HLER-EINSTEIN
METRICS ON PSEUDOCONVEX DOMAINS
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YOUNG-JUNCHOI
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Abstract. This paper is a sequel to [3] in Math. Ann. In that paper we

2 studiedthesubharmonicityofK¨ahler-Einsteinmetricsonstronglypseudocon-
1 vexdomainsofdimensiongreaterthanorequal to3. Inthispaper,westudy
thevariationsK¨ahler-Einsteinmetricsonboundedstronglypseudoconvexdo-

] mainsofdimension2. Inaddition,wediscussthepreviousresultwithgeneral
V
bounded pseudoconvex domain and local triviality of a family of bounded
C stronglypseudoconvex domains.
.
h
t 1. Introduction
a
m Let (z,s) Cn C be the standard coordinates and π : Cn C C be the
[ projection on∈the s×econd factor. Let D be a smooth domain in Cn×+1 s→uch that for

each s π(D), the slice D =D π−1(s)= z :(z,s) D is a bounded strongly
2 ∈ s ∩ { ∈ }
v pseudoconvex domian with smooth boundary.
6 In [2], Cheng and Yau constructed a unique complete K¨ahler-Einstein metric
1 on a bounded strongly pseudoconvex domain with smooth boundary. This implies
24 that there exists a unique complete K¨abler-Einstein metric hαβ¯(z,s):=hsαβ¯(z) on
each slice D which satisfies the following:
0 s
.
01 −(n+1)hαβ¯(z,s)=Ricαβ¯(z,s) (the Ricci tensor)
∂2
15 =−∂zα∂zβ¯logdet hγδ¯(z,s) 1≤γ,δ≤n.
: (cid:0) (cid:1)
v Namely, the Ricci curvature is a negative constant (n+1). This constant could
−
i be any negative number; (n+1) is chosen for convenience. On each slice D ,
X s
−
r 1
a h(z,s):= n+1logdet hγδ¯(z,s) 1≤γ,δ≤n
(cid:0) (cid:1)
isapotentialfunctionoftheK¨ahler-Einsteinmetrichαβ¯(·,s). Wecanconsiderhas
asmoothfunctiononD([3]). ItisanimmediateconsequenceoftheK¨ahler-Einstein
conditions that the restriction of h to each slice D is strictly plurisubharmonic.
s
But it is not obvious that it is also plurisubharmonic or strictly plurisubharmonic
in the base direction (the s-direction). In [3], we have shown that if the slice
dimensionnisgreaterthanorequalto3,thenhisplurisubharmonicprovidedD is
pseudoconvex. Moreover, we have also proved that h is strictly plurisubharmonic
if D is strongly pseudoconvex.
2010 Mathematics Subject Classification. 32Q20,32F05,32T15.
Key words and phrases. K¨ahler-Einstein metric, strongly pseudoconvex domain, a family of
stronglypseudoconvex domains,subharmonic,plurisubharmonic,variation.
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2 YOUNG-JUNCHOI
In this paper, we shall deal with a family of bounded smooth strongly pseudo-
convex domain of dimension greater than or equal to 2. It is remarkable to note
that Maitani and Yamaguchi already proved the 1-dimensional slice case ([10]).
Theorem 1.1. With the above notations, if D is a strongly pseudoconvex domain
in Cn+1, then h(z,s) is a strictly plurisubharmonic function.
In case of a general bounded pseudoconvex domain, Cheng and Yau also con-
structedauniqueK¨ahler-Einsteinmetricwhichisalmostcomplete,whichisalimit
of K¨ahler-Einstein metrics on relatively compact subdomains ([2]). In [8], Mok
and Yau proved that this metric is, in fact, complete. Hence we can consider the
situation that D is a pseudoconvex domain such that each slice D is a bounded
s
pseudoconvex domain. By simple approximation process, we have the following
corollary.
Corollary 1.2. Under the above hypothesis, h is a plurisubharmonic function.
In [12], Tsuji showed a dynamical construction of a K¨ahler-Eistein metric on a
boundedstronglypseudoconvexdomainwithsmoothboundary. Moreprecisely,he
have shown that the K¨ahler-Einstein metric is the iterating limit of the Bergman
metric. Using the Berndtsson’s result ([1]), he proved the same result with Corol-
lary 1.2.
The above setting is also considered as a family of bounded strongly pseudo-
convex domains. Moreover, the geodesic curvature (which is defined in Section 2)
is strongly related with the Kodaira-Spencer map. So it is natural to ask what
happens if the geodesic curvature vanishes. The following theorem answers this
question.
Theorem 1.3. Suppose that the slice dimension n is greater than or equal to 3. If
the geodesic curvature vanishes, then the family is locally trivial.
The proof of Theorem 1.3 depends the vanishing order of the solution of com-
plex Monge-Amp`ere equation near the boundary. This is why our method is not
applicable to the case that the slice dimension n=2.
Wewillallthetimeconsideronlythecaseofaonedimensionalbase,butthecom-
putationsareeasilygeneralizedtothe caseofahigherdimensionalbase. Through-
out this paper we use small Greek letters, α,β, =1,...,n for indices on z Cn
unless otherwise specified. For a properly differ·e·n·tiable function f on Cn C∈, we
×
denote by
∂f ∂f
fα = ∂zα and fβ¯ = ∂zβ¯.
where zβ¯ meanzβ. If there is no confusion, we alwaysuse the Einsteinconvention.
For a complex manifold X, we denote by T′X the complex tangent vector bundle
of X of type (1,0).
2. Prelimiaries
In this section, we recaptulate the result in [3]. Throughout this section, D is a
smooth domain in Cn+1 such that every slice
D =D π−1(s)= z :(z,s) D
s
∩ { ∈ }
SUBHARMONICITY OF KA¨HLER-EINSTEIN METRICS 3
is a bounded strongly pseudoconvex domain with smooth boundary. Since our
computation is always local in s-variable, we may assume that π(D) = U the
standard unit disc in C.
2.1. Horizontal lifts and Geodesic curvatures.
Definition 2.1. Letτ be a real(1,1)-formonD whichis positive definite oneach
slice D . We denote by v :=∂/∂s the holomorphic coordinate vector field.
s
1. A vector field v of type (1,0) is called a horizontal lift along D of v if v
τ s τ
satisfies the following:
(i) v ,w =0 for all w T′D ,
h τ iτ ∈ s
(ii) dπ(v )=v.
τ
2. The geodesic curvature c(τ) of τ is defined by the norm of v with respect
τ
to the sesquilinear form , induced by τ, namely,
h· ·iτ
c(τ)= v ,v .
h τ τiτ
Note that under the holomorphic coordinate (z,s), τ is written by
τ =√−1 τss¯ds∧ds¯+τsβ¯ds∧dzβ¯+ταs¯dzα∧ds¯+ταβ¯dzα∧dzβ¯ .
(cid:16) (cid:17)
Then the horizontal lift v and the geodesic curvature c(τ) can be written by the
τ
following:
∂ ∂
vτ = ∂s −τsβ¯τβ¯α∂zα and c(τ)=τss¯−τsβ¯τβ¯αταs¯.
Then it is well known that
τn+1 τn
(2.1) =c(τ) √ 1ds ds¯.
(n+1)! · n! ∧ − ∧
Itisremarkabletonotethatsinceτ ispositivedefinitewhenrestrictedtoD ,(2.1)
s
implies that if c(τ)>0( 0), then τ is a positive (semi-positive) real (1,1)-form.
≥
2.2. The geodesic curvatures of the real (1,1)-forms induced by defining
functions. Since every slice D is a bounded smooth strongly pseudoconvex do-
s
main, we cantake a defining function of D which satisfies the following conditions:
(i) ϕ C∞(D¯) and D = (z,s) Cn+1 :ϕ(z,s)<0 ,
∈ { ∈ }
(ii) ∂ϕ=0 on ∂D,
(iii) ϕα6β¯(·,s) >0 in D¯ and
(iv) ∂ ϕ=0 on ∂D.
z
(cid:0) 6 (cid:1)
We denote by g = log( ϕ). Then it follows that
− −
ϕαβ¯ ϕαϕβ¯
(2.2) gαβ¯ = ϕ + ϕ2
−
and the inverse is
(2.3) gβ¯α =( ϕ) ϕβ¯α+ ϕβ¯ϕα ,
− ϕ dϕ2!
−| |
where |dϕ|2 = ϕαϕβ¯gαβ¯. By some computation, we have gαβ¯gαgβ¯ ≤ 1. It follows
that gαβ¯ gives a complete K¨ahler metric on each Ds ([2]). Now we define the real
(1,1)-form G by G=√ 1∂∂¯g. A direct computation gives the following:
−
gsβ¯gβ¯α =ϕsβ¯ ϕβ¯α+ ϕϕβ¯ϕdαϕ2!+ dϕϕα2ϕsϕ.
−| | | | −
4 YOUNG-JUNCHOI
This equation shows that following proposition.
Proposition2.2([3]). Anyhorizontalliftv withrespecttoGissmoothlyextended
G
up to the boundary ∂D. Moreover, v is tangent to ∂D.
G ∂D
|
2.3. Fefferman’sapproximate solutionsand theboundarybehaviorofthe
solution of complex Monge-Amp`ere equation. Let Ω be a bounded strongly
pseudoconvexdomainwith smoothboundary. Givenasmoothfunctionζ onΩ, we
define J(ζ) by
J(ζ)=( 1)ndet ζ ζβ¯ .
− (cid:18) ζα ζαβ¯ (cid:19)
Note that if ζ >0 in Ω and g = logζ, then it is easy to show that
−
J(ζ)=e−(n+1)gdet gαβ¯ .
Consider the following problem: (cid:0) (cid:1)
J(ζ)=1 on Ω,
(2.4)
ζ =0 on ∂Ω.
In [6], Fefferman developed a formal technique to find approximate solutions of
(2.4):
Let ρ be a defining function of Ω such that dρ=0 on ∂Ω. We define recursively
6
ρ1 = ρ (J( ρ))−n+11 ,
− · −
(2.5) 1 J(ρl−1)
ρl =ρl−1 1+ − for 2 l n+1.
(n+2 l)l ≤ ≤
(cid:18) − (cid:19)
Then ρl satisfies the following properties:
(1) Every ρl is also a defining function of Ω. In particular, we may assume
that ev−ery ρl is considered as a smooth function defined on Cn.
(2) J(ρl) = 1+O(ρl) for l = 1,...,n+1, i.e., ρl is an approximate solution
| |
of order l for l =1,...,n+1.
By (2.5), we can write ρl =ηρ for some η C∞(Ω¯). Let w = log( ηρ) and
− ∈ − −
J( ηρ)=e−F. Then we have
−
det wαβ¯ =e(n+1)we−F,
and (cid:0) (cid:1)
(2.6) F = logJ( ηρ)= logJ(ρl)=O(ρl).
− − − | |
Since η is positive near ∂Ω, we know that w is strictly plurisubharmonic when
sufficiently close to the boundary and diverges on ∂Ω. By modifying w away from
∂Ω, we may assume that w is strictly plurisubharmonic on Ω. We denote it by w
and again write det wαβ¯ = e(n+1)we−F. Thus F is now a smooth function on
Ω and still satisfies that condition (2.6). Again η is understood to be a smooth
function on Ω¯ such t(cid:0)hat w(cid:1)= log( ηρ).
− −
Cheng and Yau’s theorem implies that we can solve the following equation:
det(wαβ¯+uαβ¯)=e(n+1)ueF det(wαβ¯)
(2.7) 1
c(wαβ¯)≤(wαβ¯+uαβ¯)≤c(wαβ¯).
SUBHARMONICITY OF KA¨HLER-EINSTEIN METRICS 5
Note that F =(n+1)w−logdet(wαβ¯). This implies that
Fαβ¯ =(n+1)wαβ¯+Rαβ¯,
whereRαβ¯ isthe componentofRiccicurvaturetensorofthe K¨ahlermetricwαβ¯. It
follows that wαβ¯+uαβ¯ dzαdzβ¯ is the unique complete K¨ahler-Einstein metric
in Ω. Cheng and Yau also described the boundary behavior of the solution u of
P(cid:0) (cid:1)
(2.7):
Theorem 2.3 (Simple Version [2]). Suppose that Ω is a smooth strongly pseu-
doconvex domain in Cn and ρ is a smooth defining function of Ω. Suppose that
F = ξ( ρ)k, 1 k n+1, ξ C∞(Ω¯). Suppose that u is a solution of (2.7).
− ≤ ≤ ∈
Then
Dpu (x)=O(ρa/2−p)
| | | |
wherea<min(2n+1,2k)and Dpu (x)istheEuclideanlengthofthep-thderivative
| |
of u.
Now suppose u be a solution to (2.7) with w= log( ρn+1)= log( ηρ) and
− − − −
F = logJ( ηρ). Then we know that
− −
F = logJ( ηρ)= log 1+ξ( ρ)n+1
− − − −
for some ξ C∞(Ω). Then Theorem 2.3 says t(cid:0)hat (cid:1)
∈
Dpu (x)=O(ρn+1/2−p−b)
| | | |
for b>0. In particular, we have
(2.8) uαβ¯ ≤O(|ρ|n−3/2−b)
for b>0. The above discussion(cid:12) als(cid:12)o implies that
(cid:12) (cid:12)
uαβ¯∈C∞(Ω)∩Cn−3/2−b(Ω¯),
for b>0 and 1 α,β n.
≤ ≤
3. Subharmonicity of Ka¨hler-Einstein metrics on strongly
pseudoconvex domains
Inthissection,weshalldiscussaboutTheorem1.1. Moreprecisely,wewillprove
the following:
Theorem 3.1. If every boundary point of D is a strongly pseudoconvex boundary
s
point of D, then h is strictly plurisubharmonic near D .
s
Remark 3.2. The above theorem have been already provedif the slice dimension
is greater than or equal to 3 in [3]. In fact, a little more is proved in [3]. This will
be discussed in Section 6.
3.1. The geodesic curvature from the approximate K¨ahler-Einstein met-
rics. Let D be a smooth domain in Cn+1 such that every slice D is strongly
s
pseudoconvex domain. Suppose that every boundary point of D is a strongly
s
pseudoconvex boundary point of the total space D. Then every slice Ds′ which is
sufficiently close to D has such property. Since our computation is always local
s
in s-variable, we may assume that π(D) = U and there exists a defining function
6 YOUNG-JUNCHOI
ϕ which satisfies the conditions in Subsection 2.2. By the argument in Subsection
2.3, we know that there exist approximate solutions ϕn+1(,s) such that
·
(3.1) J ϕn+1(,s) =1+O ϕ(,s)n+1 ,
· | · |
foreverys U. By (2.5)(cid:0), thereexis(cid:1)ts asmoo(cid:16)thpositivefu(cid:17)nction η onD¯ suchthat
∈
ϕn+1(,s) = η(,s)ϕ(,s). Hence η(,s)ϕ(,s) is another defining function of D
s
· − · · · ·
for each s U. We call it ψ(,s). Since every slice D is strongly pseudoconvex,
s
∈ ·
w = log( ψ) = log( ηϕ) is strictly plurisubharmonic in each slice D when
s
− − − −
sufficientlyclosetotheboundary. Itiseasytoseethatwcanbemodifiedawayfrom
∂D to a smooth function on D, which is strictly plurisubharmonic when restricted
on each slice D for s U (by shrinking U, if necessary); we again denote it by
s
w (cf, see [4]). Now let∈e−F = J( ηϕ). Then F is a smooth function on D¯ and
−
satisfies that
det wαβ¯(z,s) =e(n+1)w(z,s)e−F(z,s),
and (3.1) implies that (cid:0) (cid:1)
F(,s)=ξ(,s)ϕ(,s)n+1,
· · ·
for each s U, where ξ is a smooth function on D¯. Again η is understood to be a
smooth fun∈ction on D¯ such that w = −log(−ηϕ). So wαβ¯ = gαβ¯−(logη)αβ¯. We
can solve a family of complex Monge-Amp`ere equations:
det(wαβ¯(·,s)+uαβ¯(·,s))=eKu(·,s)eF(·,s)det(wαβ¯(·,s)),
(3.2) 1
c(wαβ¯(·,s))≤(wαβ¯(·,s)+uαβ¯(·,s))≤c(wαβ¯(·,s)).
We denote by u(,s) the solution of (3.2) for each s U. By Theorem 2.3 and
· ∈
(2.8), for eachslice D , we have the following boundary behavior ofthe solution u:
s
uαβ¯(·,s) ≤O(|ϕ(·,s)|n−3/2−b)
for b>0. (cid:12) (cid:12)
Now we define a real (cid:12)(1,1)-form(cid:12) W by W = √ 1∂∂¯w. We can write W as
−
follows:
W =√−1 wss¯ds∧ds¯+wsβ¯ds∧dzβ¯+wαs¯dzα∧ds¯+wαβ¯dzα∧dzβ¯ .
(cid:16) (cid:17)
To observe the horizontal lift v and the geodesic curvature c(W), we need to
W
compute the inverse of wαβ¯.
Lemma 3.3 ([3]). There exists a hermitian n n matrix
×
M =(Mαβ¯)∈Matn×n C∞(D¯) ,
which satisfies that (cid:0) (cid:1)
wβ¯α−gβ¯α =gβ¯γMγδ¯gδ¯α.
In particular, wβ¯α C∞(D¯) and wβ¯α =O(ϕ).
∈ | |
Withthehelpoftheabovelemma,wecanshowthatv hasthesameproperties
W
with v .
G
Proposition 3.4. Any horizontal lift v with respect to W is smoothly extended
W
up to the boundary ∂D. Moreover, v is tangent to ∂D.
W ∂D
|
SUBHARMONICITY OF KA¨HLER-EINSTEIN METRICS 7
Proof. Note that v is written by
W
vW = ∂∂s −wsβ¯wβ¯α∂∂zα.
Since w = log( ηϕ) = g logη and η is smooth up to the boundary, v is
W
− − −
smoothly extended up to the boundary. Moreover,
vW(ϕ)−vG(ϕ)=gsβ¯gβ¯αϕα−wsβ¯wβ¯αϕα
=gsβ¯(gβ¯α−wβ¯α)ϕα+(logη)sβ¯wβ¯αϕα
=gsβ¯gβ¯γMγδ¯gδ¯αϕα+(logη)sβ¯wβ¯αϕα
=O(ϕ),
| |
this completes the proof. (cid:3)
Recall that the geodesic curvature of c(W) is given by
c(W)=hvW,vWiW =wss¯−wsβ¯wβ¯αwαs¯.
Bythe definitionofLeviform,thegeodesiccurvaturec(W)iscomputedasfollows:
v ,v =√ 1∂∂¯w(v ,v )= w(v ,v )
h W WiW − W W L W W
1 1
= ψ(v ,v )+ ∂ψ(v )2.
ψL W W ψ2 | W |
−
Remark 3.5. We can observe the following:
(1) Since v is tangent to ∂D, ∂ϕ(v ) =0.
W W ∂D
|
(2) Since D is a smooth pseudoconvex domain, ψ(v ,v ) 0 on ∂D. It
W W
L ≥
follows that c(W) 0.
≥
(3) If D is strongly pseudoconvex at p ∂D , then ψ(v ,v ) > 0. It
s W W p
∈ L |
follows that
1
ψ(v ,v )
ψ(z,s)L W W |(z,s) →∞
−
as (z,s) p. In particular, c(W)(z,s) as (z,s) p.
→ →∞ →
3.2. Proof of Theorem 3.1. As we mentioned in Introduction, we denote by
hαβ¯(z,s) a unique complete K¨ahler-Einstein metric on a slice Ds. And we also
denote by a function h:D R defined by
→
1
h(z,s)= n+1logdet hγδ¯(z,s) 1≤γ,δ≤n.
Ifwedefineareal(1,1)-formH byH =√ (cid:0)1∂∂¯h,th(cid:1)enH isareal(1,1)-formonD
−
suchthat the restrictionon eachslice D is positive-definte by the K¨ahler-Einstein
s
condition. We denote by ∆ = ∆ the Laplace-Beltrami operator with respect
hαβ¯
to the K¨ahler-Einstein metric hαβ¯ on Ds. Schumacher proved that the geodesic
curvature c(H) of H satisfies a certain elliptic partial differential equation on each
slice. (For the proof, see [14] or [3].)
Theorem 3.6 ([14]). The following elliptic equation holds slicewise:
(3.3) ∆c(H)+(n+1)c(H)= ∂¯v 2.
H
−
(cid:12) (cid:12)
(cid:12) (cid:12)
8 YOUNG-JUNCHOI
From now on, we fix a slice D and we think the geodesic curvatures c(W)
s
and c(H) as functions on D . By the hypothesis, every boundary point of D
s s
is a strongly pseudoconvex boundary point of D. It follows that c(W) as
→ ∞
x ∂D by Remark 3.5. The following proposition is describe the boundary
s
→
behavior of c(H) in terms of c(W).
Proposition 3.7. The geodesic curvatures c(W) and c(H) go to infinity near the
boundary of the same order. More precisely, we have
c(H)
(3.4) (x) 1 as x ∂D .
s
c(W) → →
In the next subsection, we shall prove Proposition 3.7. In a moment, assuming
that, we want to complete the proof.
From (3.4) we know that c(H) is bounded from below. Then we can apply
the almost maximum principle due to Yau ([16]), namely, there exists a sequence
xk k∈N Ds such that
{ } ⊂
lim c(H)(x )=0, liminf∆c(H)(x ) 0, and
k k
k→∞∇ k→∞ ≥
lim c(H)(x )= inf c(H)(x).
k
k→∞ x∈Ds
It follows that
(n+1)c(H)(x ,y)= ∂¯v 2+∆c(H)(x ,y)>0.
k H k
Taking k , we have c(H) 0. (cid:12) (cid:12)
→∞ ≥ (cid:12) (cid:12)
We also know that c(H) as x ∂D by (3.4). But this prevents the func-
s
→∞ →
tionc(H)frombeingzero. Infact,accordingtoatheoremofKazdanandDeTurck
([5]), K¨ahler-Einsteinmetrics arerealanalyticonholomorphiccoordinates,andby
the Implicit Function Theorem, depend in a real-analytic way upon holomorphic
parameters. This also applies to the function c(H).
Proposition 3.8. Let ω be a K¨ahler form in Cn. Let f and g be non-negative
smooth functions on U Cn. Suppose
⊂
∆ f +Cf =g
ω
−
holds for some positive constant C. If f(0)=0, then f and g vanish identically in
a neighborhood of 0 Cn.
∈
Proof. It follows from the assumption that ψ has a local minimum at the origin,
and (3) implies that ∆ ψ(0)=0 and f(0)=0.
ωU
We set ∆ = ∆ and choose normal coordinates zα of the second kind for ω
ωU U
at 0. Let ∆ = n ∂2 be the standard Laplacian so that
0 α=1 ∂zα∂zα¯
P ∆= ∆ +tβ¯α ∂2
− − 0 ∂zα∂zβ¯
where the power series expansion of all tβ¯α have no terms of order zero or one.
Then the maximum principle of E. Hopf implies that ψ 0. (cf. See Theorem 6,
Chap. 2, Sect. 3 in [11].) ≡ (cid:3)
Therealanalyticityofc(H)andProposition3.8saythatc(H)iseitheridentically
zero, or never zero. However we know that c(W)(x) as x ∂D . This
s
→ ∞ →
completes the proof.
SUBHARMONICITY OF KA¨HLER-EINSTEIN METRICS 9
3.3. The boundary behavior of c(H). Inthis subsection,weshallprovePropo-
sition 3.7.
Recall that Remark 3.5 says that c(W) is given by the following:
1 1
c(W)= ψ(v ,v )+ ∂ψ(v )2.
ψL W W ψ2 | W |
−
Since every boundary point of D is a strongly pseudoconvexboundary point of D
s
and v is tangent to ∂D, we have
W
c(W) C ψ
≥ ·| |
for some constantC >0 when a point goes to ∂D , in particularc(W) blows up of
s
order greater than or equal to 1. To compute c(H) in terms of c(W), we need the
following lemma.
Lemma 3.9 ([3]). For each s U, there exists a hermitian n n matrix
∈ ×
Ns =(Ns ) Mat C∞(D ) Cn−3/2−b(D¯ )
αβ¯ ∈ n×n s ∩ s
(cid:16) (cid:17)
with Ns =O(ϕ(,s)n−3/2−b) for b>0, which satisfies that
k k | · |
hβ¯α(,s) wβ¯α(,s)=wβ¯γ(,s)Ns wδ¯α(,s).
· − · · γδ¯ ·
In particular, hβ¯α(,s) C∞(D ) Cn−3/2−b(D¯ ) for b > 0 and hβ¯α(,s) =
s s
· ∈ ∩ ·
O(ϕ(,s)).
| · |
By Lemma 3.9, c(H) is computed as follows:
c(H)=hss¯−hsβ¯hβ¯αhαs¯
=hss¯−hsβ¯ wβ¯α+wβ¯γNγδ¯wδ¯α hαs¯
=wss¯+uss¯(cid:16)− wsβ¯+usβ¯ wβ¯α(cid:17)+wβ¯γNγδ¯wδ¯α (wαs¯+uαs¯)
=c(W)+(rem(cid:0)aining term(cid:1)s(cid:16)), (cid:17)
where the remaining terms are given by the sum of
R1 :=uss¯+ wsβ¯wβ¯αuαs¯+usβ¯wβ¯αwαs¯+usβ¯wβ¯αuαs¯
and (cid:16) (cid:17)
R2 := wsβ¯+usβ¯ wβ¯γNγδ¯wδ¯α(wαs¯+uαs¯).
Hence it is enough to show that
(cid:0) (cid:1)
R +R
1 2
0 as x ∂D .
s
c(W) → →
First we note that u is bounded by Section 3 in [3]. By taking logarithmof (3.2)
ss¯
anddifferentiatingitwithrespecttos,weknowthatu satisfiesthefollowinglinear
s
elliptic partial differential equation on each slice D :
s
(3.5) ∆u +(n+1)u =Q,
s s
−
where Q = F + ∆ ∆ w . Here ∆ is the Laplace-Beltrami operator
− s − wαβ¯ s wαβ¯
with respect to the(cid:16)K¨ahler me(cid:17)tic wαβ¯. Note that the boundary behavior of the
solution of complex Monge-Amp`ere equation implies that
(3.6) Q=O(ϕn−3/2−b)
| |
10 YOUNG-JUNCHOI
for b>0. We need the following lemma.
Proposition 3.10. Let 0<r 1. If Q=O(ϕr), then u =O(ϕr).
s
≤ | | | | | |
Proof. In case of r = 1, it is proved in [3]. Thus we may assume that 0 < r < 1.
For c>0, we compute
∆(us−c(−ϕ)r)=hαβ¯(us−c(−ϕ)r)αβ¯
=hαβ¯(us)αβ¯−hαβ¯(c(−ϕ)r)αβ¯
=(n+1)(u ) Q hαβ¯ (cr( ϕ)r−1)( ϕ)
s − − − − α β¯
=(n+1)(us)−Q−cr(r(cid:0)−1)(−ϕ)r−2hαβ¯(−ϕ(cid:1))α(−ϕ)β¯
+cr(−ϕ)r−1hαβ¯ϕαβ¯.
Since hαβ¯ is positive definite and ϕ is plurisubharmonic, we know that
hαβ¯(−ϕ)α(−ϕ)β¯ >0 and hαβ¯ϕαβ¯ >0.
It follows that
−cr(r−1)(−ϕ)r−2hαβ¯(−ϕ)α(−ϕ)β¯+cr(−ϕ)r−1hαβ¯ϕαβ¯ >0.
So we have
∆(u c( ϕ)r) (n+1)(u ) Q.
s s
− − ≥ −
Since Q=O(ϕr), we can choose c >0 such that
1
| |
∆(u c ( ϕ)r) (n+1)(u c ( ϕ)r).
s 1 s 1
− − ≥ − −
Notethatu isboundedbySection3in[3]. ThealmostmaximumprincipleofYau
s
([16]) implies that u c ( ϕ)r 0 , i.e., u c ( ϕ)r in D .
s 1 1 s
− − ≤ ≤ −
If we apply the same argument to ∆(u +c( ϕ)r) for c>0, then we have that
s
u c (ϕr) for some constant c >0. There−fore u =O(ϕr) as desired. (cid:3)
s 2 2 s
≥− | | | |
Let(V,(v1,...,vn))beacoordinatesysteminΩsatisfyingtheconditionsinDef-
inition1.1in[2]. (cf,see[3].) Notethateveryboundedsmoothstronglypseudocon-
vexdomaininCn admits aopencoveringofsuchcoordinates. (This is constructed
in Section 1 in [2]). For a smooth function f, we write
∂|α|+|β|
f = sup f(z)
| |k+ε,V z∈V |α|+X|β|≤k(cid:12)(cid:12)∂vα∂vβ¯ (cid:12)(cid:12)
 (cid:12) (cid:12)
(cid:12) (cid:12) ∂|α|+|β| ∂|α|+|β|
+ sup z z′ −ε f(z) f(z′) ,
z,z′∈V |α|+X|β|=k| − | (cid:12)(cid:12)∂vα∂vβ¯ − ∂vα∂vβ¯ (cid:12)(cid:12)
 (cid:12) (cid:12)
where k is non-negative integer and ε (0,1)(cid:12). (cid:12)
∈
Applying the Schauder estimates to Equation (3.5) in the coordinate system
(V,(v1,...,vn)), we obtain that
(3.7) u ∗ C u + Q(2) .
| s|2+ε,V ≤ | s|0,V | |0+ε,V
(cid:16) (cid:17)
(For detailed notations, we refer to see [7].) Instead of introducing the definitions
of ∗ and (2) , we note that the construction of the coordinate system
|·|k+ε,V |·|k+ε,V